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decimal equivalents of fractions -- theory

In a ratio of integers, the corresponding decimal value either terminates or repeats in no more digits than the size of the denominator.
 
Could you please explain in detail why this is true?
 
Thank you so much in advance! I'm struggling to explain this to my daughter (we are going through a Connected Mathematics chapter together). 
 
Elena

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Andy C. | Math/Physics TutorMath/Physics Tutor
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Because not all of them will divide evenly.
 
To change a fraction into a decimal number, you divide the top number 
by the bottom number. That is, the top number goes into the division box.
You then put the decimal point after the top number and add as many
zeros on the back end as you like. Upon division, the calculations
will either stop with a remainder of zero or a repeating pattern will
occur.                                                              
                                                                   0.75
 
                                                                  _______
For example:   3/4 =   3 divided by 4  =     4 | 3.00000
                                                                     28
                                                                     --------
                                                                       20
                                                                       20
                                                                       ----
                                                                         0
 
On the other hand, the fraction 1/3 = 0.33333333.... = 0.3
 
where the bar indicates that the digits repeat
 
23/99 = 0.2323232323 = 0.23
 
The 7s are interesting creatures. They exhibit the property you are speaking of.
There are no more repeating digits than the denominator.
1/7 = 0.142857 <--- 6 repeating digits
 
This is because the remainder must be LESS than the number you are dividing by.
 
 
 

Comments

Andy, thanks so much for responding quickly! I'm still not sure I understand. I am following your first two examples (even though we write division boxes the opposite way in the Russian system ;-)) and understand that for ratios of integers, the decimals that represent them will either terminate or end in repeating patterns (otherwise, it's an irrational number).
 
I don't understand why these repeated patterns though will necessarily have fewer numbers than the size of the denominator. Is there a more theoretical proof of this? Say, when you divide 11 by 23, why would the cycle in the repeated pattern be fewer than 23 digits? What prevents it from being 25 digits long (as long as there is a cycle)?
 
Thanks so much! I really need to understand this well, before I attempt to explain it to my daughter.
We are working on Connected Math, grade 8, Looking for Pythagoras, problem 4.2 (Representing Fractions as Decimals).
 
Elena